ABSTRACT
This chapter starts from scratch, by discussing complex numbers, complex valued functions, and complex calculus, i.e. differentiation and integration, of functions in the complex plane, as well as some of the important uses of complex variable theory to fluid motion and special functions. Products of analytic functions are analytic, and quotients of analytic functions are analytic unless the denominator function is zero. The chapter provides the basis for essentially all integration in the complex plane. This is because of the remarkable fact known as the Cauchy Integral Formula. One of the most interesting classical applications of complex variable theory is to the flow of two-dimensional inviscid fluids. This subject matter is of limited usefulness for the study of real, viscous, three-dimensional flows, and in a modern context, most fluid flow problems require the use of heavy machinery.
